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Title:Annular breadth of hinges & hinge exit paths of annuli
Author(s):Tichenor, Scott R.
Director of Research:Alexander, Stephanie
Doctoral Committee Chair(s):Reznick, Bruce
Doctoral Committee Member(s):Wetzel, John E; Bishop, Richard
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):exit path
escape path
width
annulus
polygonal arc
breadth
Wetzel
broadworm
Voronoi
Rivlin
Abstract:Given a compact set $\textsf{S}\subset\mathds{R}^2$, we define the annular width function for $\textsf{S}$, denoted $w(E)$, as the width of the annulus of support of $\textsf{S}$ centered at $E\in\overline{\mathds{R}^2}$, where $\overline{\mathds{R}^2}$ is an extension of the real plane $\mathds{R}^2$. The annular breadth of $\textsf{S}$ is defined as the absolute minimum of $w(E)$. We find the $2$-segment polygonal arc with the greatest annular breadth. For a given set $\textsf{S}\subset\mathds{R}^2$, an exit path of $\textsf{S}$ is a curve that cannot be covered by the interior of $\textsf{S}$. Given an annulus, we find its shortest $1$- or $2$-segment polygonal arc exit path(s). Bezdek and Connelly provided a lengthy and technically demanding proof that \emph{All orbiforms of width} $1$ \emph{are translation covers of the set of closed planar curves of length} $2$ \emph{or less}. We provide a short and simple proof that \emph{All orbiforms of width} $1$ \emph{are covers of the set of all planar curves of length} $1$ \emph{or less}. We also provide a proof that \emph{The Reuleaux triangle of width} $1$ \emph{is a cover of the set of all closed curves of length} $2$ using a recent of Wichiramala.
Issue Date:2019-07-03
Type:Text
URI:http://hdl.handle.net/2142/105618
Rights Information:Copyright 2019 by Scott R. Tichenor. All rights reserved.
Date Available in IDEALS:2019-11-26
Date Deposited:2019-08


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